Gradient estimates and Liouville theorems for Lichnerowicz-type equation on Riemannian manifolds

In this paper we consider the gradient estimates on positive solutions to the following elliptic (Lichnerowicz) equation defined on a complete Riemannian manifold $(M,\,g)$: $$\Delta v + \mu v + a v^{p+1} +b v^{-q+1} =0,$$ where $p\geq-1$, $q\geq1$, $\mu$, $a$ and $b$ are real constants. In the case $\mu\geq0$ and $b\geq0$ or $\mu<0$ , $a>0$ and $b>0$ ($\mu$ has a lower bound), we employ the Nash-Moser iteration technique to obtain some refined gradient estimates of the solutions to the above equation, if $(M,\,g)$ satisfies $Ric \geq -(n-1)\kappa$ , where $n\geq3$ is the dimension of $M$ and $\kappa$ is a nonnegative constant, and $\mu$ , $a$ , $b$ , $p$ and $q$ satisfy some technique conditions. By the obtained gradient estimates we also derive some Liouville type theorems for the above equation under some suitable geometric and analysis conditions. As applications, we can derive some Cheng-Yau's type gradient estimates for solutions to the $n$-dimensional Einstein-scalar field Lichnerowicz equation where $n\geq3$.